Page 171 - Curvature and Homology

basis will be called a J-basis where J is the linear endomorphism defining
the complex structure of V. Any two J-bases {ei, e,.), {ei, el.} are
related by equations of the form

where (a:) is a non-singular n x n matrix with complex coefficients,
that is (a:) is an element of the general linear group GL(n, C) satisfying
a:: = a:. With respect to a J-basis the tensor J has components FBA
where
F,:
Fji*
=
F,' = F: = - ~XS:, 0. (5.2.4)
=
Hence, an element v E V (as a subset of Vc) has the components (vi, vi*)
where vi* = Ci and its image by J the components (Jv)' = dqvi,
(Jv)i* = - dqvi*.
Consider the real basis Cfi, fi,) defined in terms of the J-basis
{e,, e5*) of Vc:

Since

the vectors {fi, fi.}, i = 1, ..., n define a basis of Vc as well as V. Con-
versely, from a basis of V of the type (fi, f,,), where&, = Jf, we obtain
from (5.2.6) a basis of Vc, since gi = e,,.
If the matrix (a:) in (5.2.3) is written as (a2 = (bj) + d?(ci)
where (b;), (cj) are n x n matrices, any two real bases of the type defined
by (5.2.5) are related by a matrix of the form

called the real representation of the matrix (a:). We remark that the
determinant of the real representation of (a:) is I det(a:) I2 > 0.
With respect to the real basis Cfi, fie) the tensor J is given by the matrix